Properties

Label 2475.2.f.d
Level $2475$
Weight $2$
Character orbit 2475.f
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + \beta q^{7} -3 q^{8} +O(q^{10})\) \( q + q^{2} - q^{4} + \beta q^{7} -3 q^{8} + ( 3 + \beta ) q^{11} -2 \beta q^{13} + \beta q^{14} - q^{16} + 2 q^{17} + 3 \beta q^{19} + ( 3 + \beta ) q^{22} -2 \beta q^{26} -\beta q^{28} -6 q^{29} + 5 q^{32} + 2 q^{34} -6 q^{37} + 3 \beta q^{38} -6 q^{41} + \beta q^{43} + ( -3 - \beta ) q^{44} + 6 \beta q^{47} + 5 q^{49} + 2 \beta q^{52} + 3 \beta q^{53} -3 \beta q^{56} -6 q^{58} + 6 \beta q^{59} + 6 \beta q^{61} + 7 q^{64} -2 q^{68} + 6 \beta q^{71} + 8 \beta q^{73} -6 q^{74} -3 \beta q^{76} + ( -2 + 3 \beta ) q^{77} + 9 \beta q^{79} -6 q^{82} + 14 q^{83} + \beta q^{86} + ( -9 - 3 \beta ) q^{88} -5 \beta q^{89} + 4 q^{91} + 6 \beta q^{94} -12 q^{97} + 5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{4} - 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{4} - 6q^{8} + 6q^{11} - 2q^{16} + 4q^{17} + 6q^{22} - 12q^{29} + 10q^{32} + 4q^{34} - 12q^{37} - 12q^{41} - 6q^{44} + 10q^{49} - 12q^{58} + 14q^{64} - 4q^{68} - 12q^{74} - 4q^{77} - 12q^{82} + 28q^{83} - 18q^{88} + 8q^{91} - 24q^{97} + 10q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2276.1
1.41421i
1.41421i
1.00000 0 −1.00000 0 0 1.41421i −3.00000 0 0
2276.2 1.00000 0 −1.00000 0 0 1.41421i −3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.f.d 2
3.b odd 2 1 2475.2.f.a 2
5.b even 2 1 2475.2.f.b 2
5.c odd 4 2 495.2.d.b yes 4
11.b odd 2 1 2475.2.f.a 2
15.d odd 2 1 2475.2.f.c 2
15.e even 4 2 495.2.d.a 4
33.d even 2 1 inner 2475.2.f.d 2
55.d odd 2 1 2475.2.f.c 2
55.e even 4 2 495.2.d.a 4
165.d even 2 1 2475.2.f.b 2
165.l odd 4 2 495.2.d.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.d.a 4 15.e even 4 2
495.2.d.a 4 55.e even 4 2
495.2.d.b yes 4 5.c odd 4 2
495.2.d.b yes 4 165.l odd 4 2
2475.2.f.a 2 3.b odd 2 1
2475.2.f.a 2 11.b odd 2 1
2475.2.f.b 2 5.b even 2 1
2475.2.f.b 2 165.d even 2 1
2475.2.f.c 2 15.d odd 2 1
2475.2.f.c 2 55.d odd 2 1
2475.2.f.d 2 1.a even 1 1 trivial
2475.2.f.d 2 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2} - 1 \)
\( T_{29} + 6 \)
\( T_{37} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 2 + T^{2} \)
$11$ \( 11 - 6 T + T^{2} \)
$13$ \( 8 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 18 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 2 + T^{2} \)
$47$ \( 72 + T^{2} \)
$53$ \( 18 + T^{2} \)
$59$ \( 72 + T^{2} \)
$61$ \( 72 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( 72 + T^{2} \)
$73$ \( 128 + T^{2} \)
$79$ \( 162 + T^{2} \)
$83$ \( ( -14 + T )^{2} \)
$89$ \( 50 + T^{2} \)
$97$ \( ( 12 + T )^{2} \)
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